Abstract
The purpose of the theory of stochastic integration is to give a reasonable meaning to the idea of a differential to as wide a class of stochastic processes as possible. We saw in Sect. 7 of Chap. I that using Stieltjes integration on a path-by-path basis excludes such fundamental processes as Brownian motion, and martingales in general. Markov processes also in general have paths of unbounded variation and are similarly excluded. Therefore we must find an approach more general than Stieltjes integration.
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Bibliographic Notes
The definition of semimartingale and the treatment of stochastic integration as a Riemann-type limit of sums is in essence new. It has its origins in the fundamental theorem of Bichteler [1,2], and Dellacherie [2]. The pedagogic approach used here was first suggested by Meyer [IS], and it was then outlined by Dellacherie [2]. Dellacherie’s outline was further expanded by Lenglart [3] and Protter [6,7]. A similar idea was developed by Letta [1].
We will not attempt to give a comprehensive history of stochastic integration here, but rather just a sketch. The important early work was that of Wiener [1,2], and then of course Ito [1–5]. Doob stressed the martingale nature of the Ito integral in his book [1] and proposed a general martingale integral. Doob’s proposed development depended on a decomposition theorem (the Doob-Meyer decomposition, Theorem 6 of Chap. III) which did not yet exist. Meyer proved this decomposition theorem in [1,2], and commented that a theory of stochastic integration was now possible. This was begun by Courrège [1], and extended by Kunita and Watanabe [1], who revealed an elegant structure of square-integrable martingales and established a general change of variables formula. Meyer [4–7] extended Kunita and Watanabe’s work, realizing that the restriction of integrands to predictable processes is essential. He also extended the integrals to local martingales, which had been introduced earlier by Ito and Watanabe [1], Up to this point, stochastic integration was tied indirectly to Markov processes, by the assumption that the underlying filtration of σ-algebras be “quasi-left continuous.” This hypothesis was removed by Doléans-Dade and Meyer [1], thereby making stochastic integration a purely martingale theory. It was also in this article that semimartingales were first proposed in the form we refer to as classical semimartingales in Chap. III.
A different theory of stochastic integration was developed independently by McShane [1,2], which was close in spirit to the approach given here. However it was technically complicated and not very general. It was shown in Protter [5] (building on the work of Pop-Stojanovic [1]) that the theory of McShane could for practical purposes be viewed as a special case of the semimartingale theory.
The subject of stochastic integration essentially lay dormant for six years until Meyer [8] published a seminal “course” on stochastic integration. It was here that the importance of semimartingales was made clear, but it was not until the late 1970’s that the theorem of Bichteler [1,2], and Dellacherie [2] gave an a posteriori justification of semimartingales: The seemingly ad hoc definition of a semimartingale as a process having a decomposition into the sum of a local martingale and an FV process was shown to be the most general reasonable stochastic differential possible. (See also Kussmaul [1] in this regard, and the bibliographic notes in Chap. III.)
Most of the results of this chapter can be found in Meyer [8], though they are proven for classical semimartingales and hence of necessity the proofs are much more complicated. Theorem 4 (Stricker’s Theorem) is (of course) due to Strieker [1]; see also Meyer [10]. Theorem 5 is due to Meyer [11]. There are many other methods of expanding a filtration and still preserving the semimartingale property. The initial result is that of Ito [8], while a general reference is Jeulin [1]. A simple proof of Itò’s original result (with an extension) is in Jacod-Protter [1]. A pretty, general result is in Jacod [2].
Theorem 14 is originally due to Lenglart [1]. Theorem 16 is known to be true only in the case of integrands in L. The local behavior of the integral (Theorems 17 and 18) is due to Meyer [8] (see also McShane [1]). The a.s. Kunita-Watanabe inequality, Theorem 25, is due to Meyer [8], while the expected version (the Corollary to Theorem 25) is due to Kunita-Watanabe
That continuous martingales have paths of infinite variation or are constant a.s. was first published by Fisk [1] (Corollary 1 of Theorem 27). The proof given here of Corollary 4 of Theorem 27 (that a continuous local martingale X and its quadratic variation [X, X] have the same intervals of constancy) is due to Maisonneuve [1]. The proof of Itô’s formula (Theorem 32) is by now classic; however we benefited from Föllmer’s presentation of it [1].
The Fisk-Stratonovich integral was developed independently by Fisk [1] and Stratonovich [1], and it was extended to general semimartingales by Meyer [8]. Theorem 35 is inspired by the work of Getoor and Sharpe [1].
The stochastic exponential of Theorem 36 is due to Doléans-Dade [2]. It has become extraordinarily important. See, for example, Jacod-Shiryaev [1]. The pretty formula of Theorem 37 is due to Yor [1]. Exponentials have of course a long history in analysis. For an insightful discussion of exponentials see Gill-Johansen [1]. That every continuous local martingale is the time change of a Brownian motion is originally due to Dubins-Schwarz [1] and Dambis [1]. The proof of Lévy’s stochastic area formula (Theorem 42) is new and is due to S. Janson. See Janson-Wichura [1] for related results. The original result is in Levy [2], and another proof can be found in Yor [5].
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© 1990 Springer-Verlag Berlin Heidelberg
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Protter, P. (1990). Semimartingales and Stochastic Integrals. In: Stochastic Integration and Differential Equations. Applications of Mathematics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02619-9_3
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